Let two numbers B, C be set out which have not to one another the ratio which a square number has to a square number,
that is, which are not similar plane numbers;
and let it be contrived that, as B is to C, so is the square on A to the square on D – for we have learnt how to do this — [X. 6, Por.]
therefore the square on A is commensurable with the square on D. [X. 6]
And, since B has not to C the ratio which a square number has to a square number,
therefore neither has the square on A to the square on D the ratio which a square number has to a square number;
therefore A is incommensurable in length with D. [X. 9]
Let E be taken a mean proportional between A, D;
therefore, as A is to D, so is the square on A to the square on E. [V. Def. 9]
But A is incommensurable in length with D;
therefore the square on A is also incommensurable with the square on E; [X. 11]
therefore A is incommensurable in square with E.
Complete text
Book I